The number of bacteria P(h) in a certain population increases according to the following function, where time h is measured in hours. P(h)= 1900e^0.19h  How many hours will it take for the number of bacteria to reach 2500?

Round your answer to the nearest tenth, and do not round any intermediate computations.

To find out how many hours it will take for the number of bacteria to reach 2500, given the function:

\[
P(h) = 1900e^{0.19h}
\]

We need to solve for \( h \) when \( P(h) = 2500 \).

### Step-by-Step Solution:

1. **Set up the equation:**

   \[
   1900e^{0.19h} = 2500
   \]

2. **Divide both sides by 1900 to isolate the exponential term:**

   \[
   e^{0.19h} = \frac{2500}{1900}
   \]

   Simplifying the fraction:

   \[
   e^{0.19h} \approx 1.3158
   \]

3. **Take the natural logarithm (ln) of both sides to solve for \( h \):**

   \[
   \ln(e^{0.19h}) = \ln(1.3158)
   \]

   Since \( \ln(e^x) = x \), this simplifies to:

   \[
   0.19h = \ln(1.3158)
   \]

4. **Calculate \( \ln(1.3158) \) and then solve for \( h \):**

   \[
   \ln(1.3158) \approx 0.2746
   \]

   So:

   \[
   0.19h = 0.2746
   \]

   \[
   h = \frac{0.2746}{0.19} \approx 1.4453 \text{ hours}
   \]

5. **Round to the nearest tenth:**

   \[
   h \approx 1.4 \text{ hours}
   \]

### Final Answer:
It will take approximately **1.4 hours** for the number of bacteria to reach 2500.

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:

1. How many hours will it take for the bacteria to double in number?
2. What is the growth rate of the bacteria per hour?
3. How would the time change if the initial number of bacteria was 1000 instead of 1900?
4. What would happen to the time if the growth rate was increased to 0.25?
5. How long will it take for the bacteria to reach 5000?
6. Can you derive the equation to find the time required for any bacteria population to reach a specific number?
7. How would the time change if the growth rate were negative?
8. How does changing the base population affect the growth curve?

**Tip:** Always keep intermediate results unrounded to maintain accuracy in your final answer.

Learn how to calculate the time required for bacteria to reach a specific number using an exponential growth model. This problem involves using the exponential function P(h) = 1900 * e^(0.19h) and natural logarithms to solve for time. Suitable for advanced high school students and anyone interested in exponential growth models.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation